University of Michigan Geometry Seminar

Schedule of Talks

September 5: Special Meeting

Note Special Day! Still in 4088 East Hall, 3:10 - 4:00 PM!

Pierre Py (ENS Lyon)
Quasi-morphisms and Hamiltonian diffeomorphisms

Abstract: A quasi-morphism on a group \Gamma is a map \phi : \Gamma --> R such that there exist a constant C with | \phi(xy)-\phi(x)-\phi(y)| < C, for all x,y in \Gamma. Recently, Gambaudo and Ghys, as well as Entov and Polterovich, discovered that many groups of Hamiltonian diffeomorphisms admit non-trivial quasi-morphisms, although they do not admit any non-trivial homomorphism to R. I would like to survey and compare all these constructions.

September 11
Zhou Zhang (UM)
Kaehler-Ricci flow and complex Monge-Ampere equation

Abstract: The complex version of the Ricci flow, the Kaehler-Ricci flow, known to be equivalent to a scalar evolution equation, is relatively simple looking in comparison to the tensor evolution equation of the original Ricci flow. The equation is, indeeed, the time-dependent version of the complex Monge-Ampere equation, which determinants and so is highly non-linear. Pluripotential theory of the Monge-Ampere operator naturally arises in the study of this problem.The relation to Algebraic Geometry can be seen as the motivation and goal.

September 18
Brett Parker (MIT)
Holomorphic curves, adiabatic limits and the exploded category

Abstract: The exploded category is an extension of the smooth category with a good holomorphic curve theory in which some `degenerations' appear in smooth families. I will explain how this is useful for the study of (pseudo)holomorphic curves, and the relationship between the exploded category and tropical geometry.

Earlier preprints for background: arXiv:0705.2408 and arXiv:0706.3917.

September 25
Zuoqin Wang (MIT)
The twisted Mellin transform and spectral measures on toric varieties

Abstract: The topic of this lecture is the twisted Mellin transform. In one dimension this transform is an operator on L^2 functions on the real line which intertwines the differential operator, (d/dx)f(x) and the finite difference operator f(x+1)-f(x) and has a number of other interesting combinatorial properties as well. Its n-dimensional generalization turns out to play an important role in the asymptotics of spectral measures on toric varieties.

October 2

Special joint seminar with the AIM Seminar

Bill Allard (Duke)
On the regularity and curvature properties of level sets of minimizers for denoising models using total variation regularization

Abstract: Suppose s is a noisy grayscale image which we wish to denoise; mathematically, s will be a bounded real valued function on R^2. One way to do this is as follows. Let \gamma: R --> [0,\infty) be zero at zero, positive away from zero and convex. Let \epsilon >0 be a ``smoothing'' parameter which we will tune appropriately and let

F_\epsilon(ƒ) = \epsilon TV(ƒ) + \int \gamma(ƒ(x)-s(x)) dx

for ƒ: R^2 --> [0,\infty); here TV(ƒ) is the total variation of ƒ. Minimizers of F_\epsilon are used to denoise s.

In this talk we will describe some notable geometric properties of minimizers and will provide some interesting examples. It turns out that these results rest on the theory of area minimization which has been in the literature
for over 35 years.

October 9
Craig Sutton (Dartmouth)
Local spectral rigidity of bi-invariant metrics on compact Lie groups

Abstract: A long-standing conjecture in Riemannian geometry is that a standard sphere is uniquely determined by the spectrum of its Laplacian. Tanno has shown this to be true in dimension six and lower and has also established that in all dimensions the standard metric on $S^n$ is locally determined by its spectrum. In general, it is expected that the conjecture is also true for any symmetric space of compact type.

Outside of the CROSSes (compact rank one symmetric spaces) perhaps the most natural class of symmetric spaces of compact type are compact semisimple Lie groups equipped with the bi-invariant metric. In joint work with C. Gordon (Dartmouth) and D. Schueth (Berlin), we study the extent to which such spaces are spectrally determined. More precisely, we consider not only semisimple, but arbitrary compact Lie groups equipped with a bi-invariant metric. We determine that such metrics are spectrally isolated within the class of left-invariant metrics. Moreover, if we ignore multiplicities we see that the bi-invariant metric on a compact simple Lie group is locally determined by the first two distinct non-zero eigenvalues of its Laplacian within the class of left-invariant metrics of at most the same volume.

October 16
No Seminar: Fall Study Break! Resumes next week.

October 23
Ben Howard (UM)
The toric geometry of triangulated polygons in Euclidean space

Abstract: We investigate degenerations of the moduli space $M$ of all spatial polygons with side lengths r = (r_1,...,r_n) to toric varieties. There is one such degeneration for each triangulation D of a regular n-gon; these degenerations were discovered by Speyer and Sturmfels. Call the special toric fiber of the degeneration M_D. The triangulation determines n-3 special diagonals of a spatial n-gon, and the lengths of these diagonals form a system of commuting Hamiltonians with periodic orbits; however these Hamiltonians are only smooth where all diagonals are nonzero. Indeed their respective flows (called "bending flows" by Kapovich and Millson) do not extend continuously to the entire moduli space, and of course this is to be expected since $M$ is not generally a toric variety. However there is a continuous surjective map from M to M_D which is a symplectomorphism when restricted to the subspace U of M where all diagonals are nonvanishing. The bending flows on U then extend continuously to all of M_D. The toric varieties M_D were discovered by Kamiyama and Yoshida, however we realize $M_D$ as a special fiber of the Speyer-Sturmfels degeneration of $M$. We also prove similar results for the Grassmannian Gr_2(C^n) and relate all of this to Gelfand Tsetlin patterns. This is joint with John Millson and Chris Manon.

October 30
Gopal Prasad (UM)
Weakly commensurable arithmetic groups, lengths of closed geodesics and isospectral locally symmetric spaces

Abstract: The purpose of this talk is to give a brief description of the results obtained jointly with Andrei Rapinchuk on isospectral and length commensurable locally symmetric spaces with arithmetic fundamental groups. These results are obtained by analyzing a new notion of "weak commensurabilty" of Zariski-dense subgroups. We have obtained several results about weak commensurabilty which indicate that this notion will be useful in investigation of a variety of problems in geometry and ergodic theory. In our work we have used algebraic and transcendental number theory.

November 6
Speaker TBA
Title TBA

Abstract:

November 13
Speaker TBA
Title TBA

Abstract:

November 20
Ben Weinkove (Harvard)
Symplectic forms, Kahler metrics and the Calabi-Yau equation

Abstract: Yau's theorem on Kahler manifolds states that there exists a unique Kahler metric in every Kahler class with prescribed volume form. This has many applications in complex geometry. I will discuss results on Donaldson's program of extending the Calabi-Yau theory to symplectic manifolds. In a different direction, I will talk about the problem of existence of constant scalar curvature Kahler metrics, which can also be considered a generalization of Yau's theorem.

Here are some links to preprints:

Tosatti, V., Weinkove, B., Yau, S.T. Taming symplectic forms and the Calabi-Yau equation, http://www.arxiv.org/abs/math/0703773

Song, J., Weinkove, B. On the convergence and singularities of the J-flow with applications to the Mabuchi energy, http://www.arxiv.org/abs/math/0410418

Phong, D.H., Song, J., Sturm, J., Weinkove, B. The Moser-Trudinger inequality on Kahler-Einstein manifolds, http://www.arxiv.org/abs/math/0604076

November 27
Xiaodong Wang (MSU)
Harmonic Functions, Entropy, and a Characterization of the Hyperbolic Space

Abstract: Complete Riemannian manifolds with nonnegative Ricci curvature have been intensively studied and well understood. Riemannian manifolds with a negative lower bound for Ricci curvature are considerably more complicated and less understood. It turns out the bottom of spectrum of the Laplace operator plays an important role. I will first survey some recent results on such manifolds with positive bottom of spectrum. Then I will discuss a new rigidity theorem which characterizes hyperbolic manifolds. The proof uses ideas from potential theory and Brownian motion on Riemannian manifolds.

Note special day and room: Thursday, 3-4, 4096 EH.

Joint with Topology Seminar:

November 29
Natasa Sesum (Columbia)
Geometric Evolution Equations

Abstract: In this talk I will address asymptotics and convergence results for both the Ricci flow and the harmonic mean curvature flow equation. In the first part of the talk I will talk about stability results for the Ricci flow and the precise asymptotics of singularities of a complete Ricci flow on R^2. In the second part of the talk I will mention the short time existence result for the harmonic mean curvature flow with no assumption on a convexity of the initial data and show that it will still exist until it shrinks spherically to a point in a finite time.

December 4
Joerg Enders (MSU)

Reduced length based at singular time in the Ricci flow -
monotonicity and applications

Abstract: Quantities monotone in time are an important tool in the analysis of singularities arising in geometric evolution equations. I will discuss a generalization of a monotone quantity by Perelman along certain complete n-dimensional Ricci flows that become singular in finite time. Then I will talk about how "gradient shrinking solitons" arise in the equality case of the monotonicity and help in the understanding of singularities.

December 11
Cagatay Kutluhan (UM)
Symplectic Forms on Product 4-Manifolds

Abstract: Let M be a closed, oriented 3-manifold. It is known that if M fibers over S^1, then S^1 X M admits a symplectic form. A natural yet a more subtle question has been posed by Taubes: Is the converse true? It turns out that there might be a deep geometric reason behind a possible correspondence between 3-manifolds which fiber over S^1 and product 4-manifolds which admit symplectic structures. I will talk about the geometric significance of this problem and lay out the tools being used in an attempt to provide an affirmative answer to this question. This is a report on ongoing work on my thesis.

ARCHIVE OF PAST SEMINARS: Winter 2007.